Basis functions
Created: 05-07-2024. Last modified: 11-07-2024.
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knitr::opts_chunk$set(
message = FALSE, # Disable messages printed by R code chunks
warning = FALSE, # Disable warnings printed by R code chunks
echo = TRUE, # Show R code within code chunks in output
include = TRUE, # Include both R code and its results in output
eval = TRUE, # Evaluate R code chunks
cache = FALSE, # Enable caching of R code chunks for faster rendering
fig.align = "center",
out.width = "100%",
retina = 2,
error = TRUE,
collapse = FALSE
)
rm(list = ls())
set.seed(1982)1 Import libraries
2 Building the graph object
3 Geodesic distance on a metric graph
Figure 1 below shows an illustration of a compact and connected graph \(\Gamma = (\mathcal{V},\mathcal{E})\), where \(\mathcal{E}= \{e_1,\dots,e_8\}\) (black lines) and \(\mathcal{V} = \{v_1,\dots,v_7\}\) (blue points).
Press the Show button below to reveal the code.
df = data.frame(x = c(0.5,0), y = c(-1,-1), z = c(0,0))
df2 = data.frame(x = c(0,0), y = c(-1,0), z = c(0,0))
df3 = data.frame(x = c(0,-0.5), y = c(0,0), z = c(0,0))
graph$build_mesh(h = 1/40)
x <- graph$mesh$V[, 1]
y <- graph$mesh$V[, 2]
f <- x^2 - y^2
graph$plot(plotly = TRUE, vertex_size = 5, vertex_color = "blue",
edge_color = "black", edge_width = 2) %>%
config(mathjax = 'cdn') %>%
layout(title = TeX("\\text{Geodesic distance on a metric graph }\\Gamma = (\\mathcal{V},\\mathcal{E})"),
showlegend = FALSE,
font = list(family = "Palatino"),
scene = list(
aspectratio = list(x = 1.6, y = 1.6, z = 1),
annotations = list(
list(
x = -1, y = 0, z = 0,
text = TeX("v_1"),
textangle = 0, ax = 0, ay = -25,
font = list(color = "black", size = 20),
arrowcolor = "black", arrowsize = 1, arrowwidth = 1,arrowhead = 1),
list(
x = 0, y = 0, z = 0,
text = TeX("v_2"),
textangle = 0, ax = 0, ay = -25,
font = list(color = "black", size = 20),
arrowcolor = "black", arrowsize = 1, arrowwidth = 1,arrowhead = 1),
list(
x = 0, y = -1, z = 0,
text = TeX("v_3"),
textangle = 0, ax = 0, ay = -25,
font = list(color = "black", size = 20),
arrowcolor = "black", arrowsize = 1, arrowwidth = 1,arrowhead = 1),
list(
x = 1, y = -1, z = 0,
text = TeX("v_4"),
textangle = 0, ax = 0, ay = -25,
font = list(color = "black", size = 20),
arrowcolor = "black", arrowsize = 1, arrowwidth = 1,arrowhead = 1),
list(
x = 1, y = 0, z = 0,
text = TeX("v_5"),
textangle = 0, ax = 0, ay = -25,
font = list(color = "black", size = 20),
arrowcolor = "black", arrowsize = 1, arrowwidth = 1,arrowhead = 1),
list(
x = 1, y = 1, z = 0,
text = TeX("v_6"),
textangle = 0, ax = 0, ay = -25,
font = list(color = "black", size = 20),
arrowcolor = "black", arrowsize = 1, arrowwidth = 1,arrowhead = 1),
list(
x = 0, y = 1, z = 0,
text = TeX("v_7"),
textangle = 0, ax = 0, ay = -25,
font = list(color = "black", size = 20),
arrowcolor = "black", arrowsize = 1, arrowwidth = 1,arrowhead = 1),
list(
x = 0.5, y = -1, z = 0,
text = TeX("s_1"),
textangle = 0, ax = 0, ay = -25,
font = list(color = "black", size = 20),
arrowcolor = "black", arrowsize = 1, arrowwidth = 1,arrowhead = 1),
list(
x = -0.5, y = 0, z = 0,
text = TeX("s_2"),
textangle = 0, ax = 0, ay = -25,
font = list(color = "black", size = 20),
arrowcolor = "black", arrowsize = 1, arrowwidth = 1,arrowhead = 1),
list(
x = -0.5, y = 0, z = 0,
text = TeX("e_1"),
textangle = 0, ax = 0, ay = 10,
font = list(color = "black", size = 20),
arrowcolor = "white", arrowsize = 1, arrowwidth = 1,arrowhead = 1),
list(
x = 0, y = -0.5, z = 0,
text = TeX("e_2"),
textangle = 0, ax = 0, ay = 10,
font = list(color = "black", size = 20),
arrowcolor = "white", arrowsize = 1, arrowwidth = 1,arrowhead = 1),
list(
x = 0.5, y = -1, z = 0,
text = TeX("e_3"),
textangle = 0, ax = 0, ay = 10,
font = list(color = "black", size = 20),
arrowcolor = "white", arrowsize = 1, arrowwidth = 1,arrowhead = 1),
list(
x = 1, y = -0.5, z = 0,
text = TeX("e_4"),
textangle = 0, ax = 0, ay = 10,
font = list(color = "black", size = 20),
arrowcolor = "white", arrowsize = 1, arrowwidth = 1,arrowhead = 1),
list(
x = 0.5, y = 0, z = 0,
text = TeX("e_5"),
textangle = 0, ax = 0, ay = 10,
font = list(color = "black", size = 20),
arrowcolor = "white", arrowsize = 1, arrowwidth = 1,arrowhead = 1),
list(
x = 1, y = 0.5, z = 0,
text = TeX("e_6"),
textangle = 0, ax = 0, ay = 10,
font = list(color = "black", size = 20),
arrowcolor = "white", arrowsize = 1, arrowwidth = 1,arrowhead = 1),
list(
x = 0.5, y = 1, z = 0,
text = TeX("e_7"),
textangle = 0, ax = 0, ay = 10,
font = list(color = "black", size = 20),
arrowcolor = "white", arrowsize = 1, arrowwidth = 1,arrowhead = 1),
list(
x = 0, y = 0.5, z = 0,
text = TeX("e_8"),
textangle = 0, ax = 0, ay = 10,
font = list(color = "black", size = 20),
arrowcolor = "white", arrowsize = 1, arrowwidth = 1,arrowhead = 1)
))) %>%
add_trace(x = 0.5, y = -1, z = 0, mode = "markers", type = "scatter3d",
marker = list(size = 5, color = "red", symbol = 104)) %>%
add_trace(x = -0.5, y = 0, z = 0, mode = "markers", type = "scatter3d",
marker = list(size = 5, color = "red", symbol = 104)) %>%
add_trace(data = df, x = ~x, y = ~y, z = ~z, mode = "lines", type = "scatter3d",
line = list(width = 2, color = "red", symbol = 104)) %>%
add_trace(data = df2, x = ~x, y = ~y, z = ~z, mode = "lines", type = "scatter3d",
line = list(width = 2, color = "red", symbol = 104)) %>%
add_trace(data = df3, x = ~x, y = ~y, z = ~z, mode = "lines", type = "scatter3d",
line = list(width = 2, color = "red", symbol = 104))Figure 1. Illustration of a compact and connected graph \(\Gamma = \left(\mathcal{V},\mathcal{E}\right)\), where \(\mathcal{E}= \left\{e_1,\dots,e_8\right\}\) (black) and \(\mathcal{V}= \left\{v_1,\dots,v_7\right\}\) (blue). Notice that \(\ell_{e_i} = 1\) for \(i = 1,\dots,8\). Note also that, for example, \(\mathcal{E}_{v_2} = \left\{e_1,e_2,e_5,e_8\right\}\) and therefore \(d_{v_2} = 4.\) Given two points \(s_1 = (e_3, 0.5)\) and \(s_2 = (e_1, 0.5)\), we have that \(d(s_1,s_2) = 2\) (red). Following the convention mentioned in the text, we may write, for example, \(e_8 = (v_2,v_7) \cong [0,\ell_{e_8}]\).
4 Function on a metric graph
Figure 2 below provides an illustration of a function defined on a graph.
Press the Show button below to reveal the code.
graph$build_mesh(h = 1/40)
x <- graph$mesh$V[, 1]
y <- graph$mesh$V[, 2]
f <- x^2 - y^2
df = data.frame(x = c(0.5,0), y = c(-1,-1), z = c(0,0))
df2 = data.frame(x = c(0,0), y = c(-1,0), z = c(0,0))
df3 = data.frame(x = c(0,-0.5), y = c(0,0), z = c(0,0))
graph$plot_function(f,plotly = TRUE, vertex_size = 5, vertex_color = "blue", line_color = "blue",
edge_color = "black", edge_width = 2) %>%
config(mathjax = 'cdn') %>%
layout(title = TeX("\\text{Function on a metric graph }\\Gamma = (\\mathcal{V},\\mathcal{E})"),
showlegend = FALSE,
font = list(family = "Palatino"),
scene = list(
aspectratio = list(x = 1.6, y = 1.6, z = 1),
annotations = list(
list(
x = -1, y = 0, z = 0,
text = TeX("v_1"),
textangle = 0, ax = 0, ay = -25,
font = list(color = "black", size = 16),
arrowcolor = "black", arrowsize = 1, arrowwidth = 1,arrowhead = 1),
list(
x = 0, y = 0, z = 0,
text = TeX("v_2"),
textangle = 0, ax = 0, ay = -25,
font = list(color = "black", size = 16),
arrowcolor = "black", arrowsize = 1, arrowwidth = 1,arrowhead = 1),
list(
x = 0, y = -1, z = 0,
text = TeX("v_3"),
textangle = 0, ax = 0, ay = -25,
font = list(color = "black", size = 16),
arrowcolor = "black", arrowsize = 1, arrowwidth = 1,arrowhead = 1),
list(
x = 1, y = -1, z = 0,
text = TeX("v_4"),
textangle = 0, ax = 0, ay = -25,
font = list(color = "black", size = 16),
arrowcolor = "black", arrowsize = 1, arrowwidth = 1,arrowhead = 1),
list(
x = 1, y = 0, z = 0,
text = TeX("v_5"),
textangle = 0, ax = 0, ay = -25,
font = list(color = "black", size = 16),
arrowcolor = "black", arrowsize = 1, arrowwidth = 1,arrowhead = 1),
list(
x = 1, y = 1, z = 0,
text = TeX("v_6"),
textangle = 0, ax = 0, ay = -25,
font = list(color = "black", size = 16),
arrowcolor = "black", arrowsize = 1, arrowwidth = 1,arrowhead = 1),
list(
x = 0, y = 1, z = 0,
text = TeX("v_7"),
textangle = 0, ax = 0, ay = -25,
font = list(color = "black", size = 16),
arrowcolor = "black", arrowsize = 1, arrowwidth = 1,arrowhead = 1),
list(
x = 0.5, y = -1, z = 0,
text = TeX("s_1"),
textangle = 0, ax = 0, ay = -25,
font = list(color = "black", size = 16),
arrowcolor = "black", arrowsize = 1, arrowwidth = 1,arrowhead = 1),
list(
x = -0.5, y = 0, z = 0,
text = TeX("s_2"),
textangle = 0, ax = 0, ay = -25,
font = list(color = "black", size = 16),
arrowcolor = "black", arrowsize = 1, arrowwidth = 1,arrowhead = 1),
list(
x = -0.5, y = 0, z = 0,
text = TeX("e_1"),
textangle = 0, ax = 0, ay = 10,
font = list(color = "black", size = 16),
arrowcolor = "white", arrowsize = 1, arrowwidth = 1,arrowhead = 1),
list(
x = 0, y = -0.5, z = 0,
text = TeX("e_2"),
textangle = 0, ax = 0, ay = 10,
font = list(color = "black", size = 16),
arrowcolor = "white", arrowsize = 1, arrowwidth = 1,arrowhead = 1),
list(
x = 0.5, y = -1, z = 0,
text = TeX("e_3"),
textangle = 0, ax = 0, ay = 10,
font = list(color = "black", size = 16),
arrowcolor = "white", arrowsize = 1, arrowwidth = 1,arrowhead = 1),
list(
x = 1, y = -0.5, z = 0,
text = TeX("e_4"),
textangle = 0, ax = 0, ay = 10,
font = list(color = "black", size = 16),
arrowcolor = "white", arrowsize = 1, arrowwidth = 1,arrowhead = 1),
list(
x = 0.5, y = 0, z = 0,
text = TeX("e_5"),
textangle = 0, ax = 0, ay = 10,
font = list(color = "black", size = 16),
arrowcolor = "white", arrowsize = 1, arrowwidth = 1,arrowhead = 1),
list(
x = 1, y = 0.5, z = 0,
text = TeX("e_6"),
textangle = 0, ax = 0, ay = 10,
font = list(color = "black", size = 16),
arrowcolor = "white", arrowsize = 1, arrowwidth = 1,arrowhead = 1),
list(
x = 0.5, y = 1, z = 0,
text = TeX("e_7"),
textangle = 0, ax = 0, ay = 10,
font = list(color = "black", size = 16),
arrowcolor = "white", arrowsize = 1, arrowwidth = 1,arrowhead = 1),
list(
x = 0, y = 0.5, z = 0,
text = TeX("e_8"),
textangle = 0, ax = 0, ay = 10,
font = list(color = "black", size = 16),
arrowcolor = "white", arrowsize = 1, arrowwidth = 1,arrowhead = 1),
list(
x = 0, y = -1, z = 1,
text = TeX("f=\\gamma\\big|_\\Gamma"),
textangle = 0, ax = 60, ay = 0,
font = list(color = "blue", size = 16),
arrowcolor = "white", arrowsize = 1, arrowwidth = 1,arrowhead = 1)
))) %>%
add_trace(x = 0.5, y = -1, z = 0, mode = "markers", type = "scatter3d",
marker = list(size = 5, color = "red", symbol = 104)) %>%
add_trace(x = -0.5, y = 0, z = 0, mode = "markers", type = "scatter3d",
marker = list(size = 5, color = "red", symbol = 104)) %>%
add_trace(data = df, x = ~x, y = ~y, z = ~z, mode = "lines", type = "scatter3d",
line = list(width = 2, color = "red", symbol = 104)) %>%
add_trace(data = df2, x = ~x, y = ~y, z = ~z, mode = "lines", type = "scatter3d",
line = list(width = 2, color = "red", symbol = 104)) %>%
add_trace(data = df3, x = ~x, y = ~y, z = ~z, mode = "lines", type = "scatter3d",
line = list(width = 2, color = "red", symbol = 104))Figure 2. Illustration of a function \(f\) (in blue) defined on the graph \(\Gamma= (\mathcal{V},\mathcal{E})\), where \(\mathcal{E}= \left\{e_1,\dots,e_8\right\}\) and \(\mathcal{V}= \left\{v_1,\dots,v_7\right\}\). Function \(f:\Gamma\longrightarrow\mathbb{R}\) is the restriction of the surface \(\gamma:\mathbb{R}^2\longrightarrow\mathbb{R}\) to \(\Gamma\), where \(\gamma(x,y) = x^2-y^2\). That is, \(f= \gamma\big|_\Gamma\).
5 Basis functions on a metric graph
Let each edge \(e\in\mathcal{E}\) be subdivided into \(n_{e}\geq 2\) regular segments of length \(h_{e}\), and be delimited by the nodes \(0 = x_0^{e},x_1^{e},\dots,x_{n_{e}-1}^{e}, x_{n_{e}}^{e} = \ell_{e}\). For each \(j = 1,\dots,n_{e}-1\), we consider the following standard hat basis functions \[\begin{equation*} \varphi_j^{e}(x)=\begin{cases} 1-\dfrac{|x_j^{e}-x|}{h_{e}},&\text{ if }x_{j-1}^{e}\leq x\leq x_{j+1}^{e},\\ 0,&\text{ otherwise}. \end{cases} \end{equation*}\] For each \(e\in\mathcal{E}\), the set of hat functions \(\left\{\varphi_1^{e},\dots,\varphi_{n_{e}-1}^{e}\right\}\) is a basis for the space \[\begin{equation*} V_{h_{e}} = \left\{w\in H_0^1(e)\;\Big|\;\forall j = 0,1,\dots,n_{e}-1:w|_{[x_j^{e}, x_{j+1}^{e}]}\in\mathbb{P}^1\right\}, \end{equation*}\] where \(\mathbb{P}^1\) is the space of linear functions on \([0,\ell_{e}]\). For each vertex \(v\in\mathcal{V}\), we define \[\begin{equation*} \mathcal{N}_v = \left\{\bigcup_{e\in\left\{e\in\mathcal{E}_v: v = x_0^e\right\}}[v,x_1^e]\right\}\bigcup\left\{\bigcup_{e\in\left\{e\in\mathcal{E}_v: v = x^e_{n_e}\right\}}[x^e_{n_e-1},v]\right\}, \end{equation*}\] which is a star-shaped set with center at \(v\) and rays made of the segments contiguous to \(v\). On \(\mathcal{N}_v\), we define the hat functions as \[\begin{equation*} \phi_v(x)=\begin{cases} 1-\dfrac{|x_v^{e}-x|}{h_{e}},&\text{ if }x\in\mathcal{N}_v\cap e \text{ and }e\in\mathcal{E}_v,\\ 0,&\text{ otherwise}, \end{cases} \end{equation*}\] where \(x_v^e\) is either \(x_0^e\) or \(x_{n_e}^e\) depending on the edge direction and its parameterization. See Arioli and Benzi (2018) for more. Figure 3 below provides an illustration of the system of basis functions \(\{\varphi_j^e, \phi_v\}\) together with the set \(\mathcal{N}_v\).
Press the Show button below to reveal the code.
#vertical
df01 = data.frame(x = c(-1,-1), y = c(0,0), z = c(0,1))
df02 = data.frame(x = c(-2/3,-2/3), y = c(0,0), z = c(0,1))
df03 = data.frame(x = c(-1/3,-1/3), y = c(0,0), z = c(0,1))
df04 = data.frame(x = c(0,0), y = c(0,0), z = c(0,1))
df05 = data.frame(x = c(0,0), y = c(-1/3,-1/3), z = c(0,1))
df06 = data.frame(x = c(0,0), y = c(-2/3,-2/3), z = c(0,1))
df07 = data.frame(x = c(0,0), y = c(-1,-1), z = c(0,1))
df08 = data.frame(x = c(2/3,2/3), y = c(-1,-1), z = c(0,1))
df09 = data.frame(x = c(1/3,1/3), y = c(-1,-1), z = c(0,1))
df10 = data.frame(x = c(1,1), y = c(-1,-1), z = c(0,1))
df11 = data.frame(x = c(2/3,2/3), y = c(0,0), z = c(0,1))
df12 = data.frame(x = c(1/3,1/3), y = c(0,0), z = c(0,1))
df13 = data.frame(x = c(1,1), y = c(0,0), z = c(0,1))
df14 = data.frame(x = c(2/3,2/3), y = c(1,1), z = c(0,1))
df15 = data.frame(x = c(1/3,1/3), y = c(1,1), z = c(0,1))
df16 = data.frame(x = c(1,1), y = c(1,1), z = c(0,1))
df17 = data.frame(x = c(1,1), y = c(-1/3,-1/3), z = c(0,1))
df18 = data.frame(x = c(1,1), y = c(-2/3,-2/3), z = c(0,1))
df19 = data.frame(x = c(1,1), y = c(1/3,1/3), z = c(0,1))
df20 = data.frame(x = c(1,1), y = c(2/3,2/3), z = c(0,1))
df21 = data.frame(x = c(0,0), y = c(1/3,1/3), z = c(0,1))
df22 = data.frame(x = c(0,0), y = c(2/3,2/3), z = c(0,1))
df23 = data.frame(x = c(0,0), y = c(1,1), z = c(0,1))
#inclined
d01 = data.frame(x = c(-1,-2/3), y = c(0,0), z = c(0,1))
d02 = data.frame(x = c(-2/3,-1/3), y = c(0,0), z = c(0,1))
d03 = data.frame(x = c(-1/3,0), y = c(0,0), z = c(0,1)) #central
d04 = data.frame(x = c(0,1/3), y = c(0,0), z = c(0,1))
d05 = data.frame(x = c(1/3,2/3), y = c(0,0), z = c(0,1))
d06 = data.frame(x = c(2/3,1), y = c(0,0), z = c(0,1))
d07 = data.frame(x = c(1,2/3), y = c(0,0), z = c(0,1))
d08 = data.frame(x = c(2/3,1/3), y = c(0,0), z = c(0,1))
d09 = data.frame(x = c(1/3,0), y = c(0,0), z = c(0,1)) #central
d10 = data.frame(x = c(0,-1/3), y = c(0,0), z = c(0,1))
d11 = data.frame(x = c(-1/3,-2/3), y = c(0,0), z = c(0,1))
d12 = data.frame(x = c(-2/3,-1), y = c(0,0), z = c(0,1))
d13 = data.frame(x = c(0,1/3), y = c(1,1), z = c(0,1))
d14 = data.frame(x = c(1/3,2/3), y = c(1,1), z = c(0,1))#basis
d15 = data.frame(x = c(2/3,1), y = c(1,1), z = c(0,1))
d16 = data.frame(x = c(1,2/3), y = c(1,1), z = c(0,1))#basis
d17 = data.frame(x = c(2/3,1/3), y = c(1,1), z = c(0,1))
d18 = data.frame(x = c(1/3,0), y = c(1,1), z = c(0,1))
d19 = data.frame(x = c(0,1/3), y = c(-1,-1), z = c(0,1))
d20 = data.frame(x = c(1/3,2/3), y = c(-1,-1), z = c(0,1))
d21 = data.frame(x = c(2/3,1), y = c(-1,-1), z = c(0,1))
d22 = data.frame(x = c(1,2/3), y = c(-1,-1), z = c(0,1))
d23 = data.frame(x = c(2/3,1/3), y = c(-1,-1), z = c(0,1))
d24 = data.frame(x = c(1/3,0), y = c(-1,-1), z = c(0,1))
d25 = data.frame(y = c(-1,-2/3), x = c(0,0), z = c(0,1))
d26 = data.frame(y = c(-2/3,-1/3), x = c(0,0), z = c(0,1))
d27 = data.frame(y = c(-1/3,0), x = c(0,0), z = c(0,1)) #central
d28 = data.frame(y = c(0,1/3), x = c(0,0), z = c(0,1))
d29 = data.frame(y = c(1/3,2/3), x = c(0,0), z = c(0,1))
d30 = data.frame(y = c(2/3,1), x = c(0,0), z = c(0,1))
d31 = data.frame(y = c(1,2/3), x = c(0,0), z = c(0,1))
d32 = data.frame(y = c(2/3,1/3), x = c(0,0), z = c(0,1))
d33 = data.frame(y = c(1/3,0), x = c(0,0), z = c(0,1)) #central
d34 = data.frame(y = c(0,-1/3), x = c(0,0), z = c(0,1))
d35 = data.frame(y = c(-1/3,-2/3), x = c(0,0), z = c(0,1))
d36 = data.frame(y = c(-2/3,-1), x = c(0,0), z = c(0,1))
d37 = data.frame(y = c(-1,-2/3), x = c(1,1), z = c(0,1))
d38 = data.frame(y = c(-2/3,-1/3), x = c(1,1), z = c(0,1))
d39 = data.frame(y = c(-1/3,0), x = c(1,1), z = c(0,1))
d40 = data.frame(y = c(0,1/3), x = c(1,1), z = c(0,1))
d41 = data.frame(y = c(1/3,2/3), x = c(1,1), z = c(0,1))
d42 = data.frame(y = c(2/3,1), x = c(1,1), z = c(0,1))
d43 = data.frame(y = c(1,2/3), x = c(1,1), z = c(0,1))
d44 = data.frame(y = c(2/3,1/3), x = c(1,1), z = c(0,1))
d45 = data.frame(y = c(1/3,0), x = c(1,1), z = c(0,1))
d46 = data.frame(y = c(0,-1/3), x = c(1,1), z = c(0,1))
d47 = data.frame(y = c(-1/3,-2/3), x = c(1,1), z = c(0,1))
d48 = data.frame(y = c(-2/3,-1), x = c(1,1), z = c(0,1))
# star set
d49 = data.frame(y = c(1/3,0), x = c(0,0), z = c(0,0))
d50 = data.frame(y = c(0,0), x = c(1/3,0), z = c(0,0))
d51 = data.frame(y = c(-1/3,0), x = c(0,0), z = c(0,0))
d52 = data.frame(y = c(0,0), x = c(-1/3,0), z = c(0,0))
graph$build_mesh(h = 1/40)
graph$plot(plotly = TRUE, vertex_size = 4, vertex_color = "blue",
edge_color = "black", edge_width = 2) %>%
config(mathjax = 'cdn') %>%
layout(title = TeX("\\text{Basis functions on a metric graph }\\Gamma = (\\mathcal{V},\\mathcal{E})"),
showlegend = FALSE,
font = list(family = "Palatino"),
scene = list(
aspectratio = list(x = 1.8, y = 1.8, z = 0.4),
annotations = list(
list(
x = -1, y = 0, z = 0,
text = TeX("v_1"),
textangle = 0, ax = 0, ay = 25,
font = list(color = "black", size = 16),
arrowcolor = "black", arrowsize = 1, arrowwidth = 1,arrowhead = 1),
list(
x = 0, y = 0, z = 0,
text = TeX("v_2"),
textangle = 0, ax = 0, ay = 25,
font = list(color = "black", size = 16),
arrowcolor = "black", arrowsize = 1, arrowwidth = 1,arrowhead = 1),
list(
x = 0, y = -1, z = 0,
text = TeX("v_3"),
textangle = 0, ax = 0, ay = 25,
font = list(color = "black", size = 16),
arrowcolor = "black", arrowsize = 1, arrowwidth = 1,arrowhead = 1),
list(
x = 1, y = -1, z = 0,
text = TeX("v_4"),
textangle = 0, ax = 0, ay = 25,
font = list(color = "black", size = 16),
arrowcolor = "black", arrowsize = 1, arrowwidth = 1,arrowhead = 1),
list(
x = 1, y = 0, z = 0,
text = TeX("v_5"),
textangle = 0, ax = 0, ay = 25,
font = list(color = "black", size = 16),
arrowcolor = "black", arrowsize = 1, arrowwidth = 1,arrowhead = 1),
list(
x = 1, y = 1, z = 0,
text = TeX("v_6,x_0^{e_7}"),
textangle = 0, ax = 0, ay = 25,
font = list(color = "black", size = 16),
arrowcolor = "black", arrowsize = 1, arrowwidth = 1,arrowhead = 1),
list(
x = 0, y = 1, z = 0,
text = TeX("v_7,x_3^{e_7}"),
textangle = 0, ax = 0, ay = 25,
font = list(color = "black", size = 16),
arrowcolor = "black", arrowsize = 1, arrowwidth = 1,arrowhead = 1),
list(
x = 2/3, y = 1, z = 0,
text = TeX("x_1^{e_7}"),
textangle = 0, ax = 0, ay = 25,
font = list(color = "black", size = 16),
arrowcolor = "black", arrowsize = 1, arrowwidth = 1,arrowhead = 1),
list(
x = 1/3, y = 1, z = 0,
text = TeX("x_2^{e_7}"),
textangle = 0, ax = 0, ay = 25,
font = list(color = "black", size = 16),
arrowcolor = "black", arrowsize = 1, arrowwidth = 1,arrowhead = 1),
list(
x = 2/3, y = 1, z = 1,
text = TeX("\\varphi_1^{e_7}"),
textangle = 0, ax = 0, ay = 75,
font = list(color = "blue", size = 16),
arrowcolor = "white", arrowsize = 1, arrowwidth = 1,arrowhead = 1),
list(
x = 0, y = 0, z = 1,
text = TeX("\\phi_{v_2}"),
textangle = 0, ax = 0, ay = 75,
font = list(color = "red", size = 16),
arrowcolor = "white", arrowsize = 1, arrowwidth = 1,arrowhead = 1)
))) %>%
add_trace(x = -2/3, y = 0, z = 0, mode = "markers", type = "scatter3d",
marker = list(size = 4, color = "black", symbol = 104)) %>%
add_trace(x = -1/3, y = 0, z = 0, mode = "markers", type = "scatter3d",
marker = list(size = 4, color = "black", symbol = 104)) %>%
add_trace(x = 0, y = -1/3, z = 0, mode = "markers", type = "scatter3d",
marker = list(size = 4, color = "black", symbol = 104)) %>%
add_trace(x = 0, y = -2/3, z = 0, mode = "markers", type = "scatter3d",
marker = list(size = 4, color = "black", symbol = 104)) %>%
add_trace(x = 1/3, y = -1, z = 0, mode = "markers", type = "scatter3d",
marker = list(size = 4, color = "black", symbol = 104)) %>%
add_trace(x = 2/3, y = -1, z = 0, mode = "markers", type = "scatter3d",
marker = list(size = 4, color = "black", symbol = 104)) %>%
add_trace(x = 0, y = 1/3, z = 0, mode = "markers", type = "scatter3d",
marker = list(size = 4, color = "black", symbol = 104)) %>%
add_trace(x = 0, y = 2/3, z = 0, mode = "markers", type = "scatter3d",
marker = list(size = 4, color = "black", symbol = 104)) %>%
add_trace(x = 1/3, y = 0, z = 0, mode = "markers", type = "scatter3d",
marker = list(size = 4, color = "black", symbol = 104)) %>%
add_trace(x = 2/3, y = 0, z = 0, mode = "markers", type = "scatter3d",
marker = list(size = 4, color = "black", symbol = 104)) %>%
add_trace(x = 1/3, y = 1, z = 0, mode = "markers", type = "scatter3d",
marker = list(size = 4, color = "black", symbol = 104)) %>%
add_trace(x = 2/3, y = 1, z = 0, mode = "markers", type = "scatter3d",
marker = list(size = 4, color = "black", symbol = 104)) %>%
add_trace(x = 1, y = 1/3, z = 0, mode = "markers", type = "scatter3d",
marker = list(size = 4, color = "black", symbol = 104)) %>%
add_trace(x = 1, y = 2/3, z = 0, mode = "markers", type = "scatter3d",
marker = list(size = 4, color = "black", symbol = 104)) %>%
add_trace(x = 1, y = -1/3, z = 0, mode = "markers", type = "scatter3d",
marker = list(size = 4, color = "black", symbol = 104)) %>%
add_trace(x = 1, y = -2/3, z = 0, mode = "markers", type = "scatter3d",
marker = list(size = 4, color = "black", symbol = 104)) %>% #so far the botton
add_trace(x = -2/3, y = 0, z = 1, mode = "markers", type = "scatter3d",
marker = list(size = 1, color = "black", symbol = 104)) %>%
add_trace(x = -1/3, y = 0, z = 1, mode = "markers", type = "scatter3d",
marker = list(size = 1, color = "black", symbol = 104)) %>%
add_trace(x = 0, y = -1/3, z = 1, mode = "markers", type = "scatter3d",
marker = list(size = 1, color = "black", symbol = 104)) %>%
add_trace(x = 0, y = -2/3, z = 1, mode = "markers", type = "scatter3d",
marker = list(size = 1, color = "black", symbol = 104)) %>%
add_trace(x = 1/3, y = -1, z = 1, mode = "markers", type = "scatter3d",
marker = list(size = 1, color = "black", symbol = 104)) %>%
add_trace(x = 2/3, y = -1, z = 1, mode = "markers", type = "scatter3d",
marker = list(size = 1, color = "black", symbol = 104)) %>%
add_trace(x = 0, y = 1/3, z = 1, mode = "markers", type = "scatter3d",
marker = list(size = 1, color = "black", symbol = 104)) %>%
add_trace(x = 0, y = 2/3, z = 1, mode = "markers", type = "scatter3d",
marker = list(size = 1, color = "black", symbol = 104)) %>%
add_trace(x = 1/3, y = 0, z = 1, mode = "markers", type = "scatter3d",
marker = list(size = 1, color = "black", symbol = 104)) %>%
add_trace(x = 2/3, y = 0, z = 1, mode = "markers", type = "scatter3d",
marker = list(size = 1, color = "black", symbol = 104)) %>%
add_trace(x = 1/3, y = 1, z = 1, mode = "markers", type = "scatter3d",
marker = list(size = 1, color = "black", symbol = 104)) %>%
add_trace(x = 2/3, y = 1, z = 1, mode = "markers", type = "scatter3d",
marker = list(size = 1, color = "black", symbol = 104)) %>%
add_trace(x = 1, y = 1/3, z = 1, mode = "markers", type = "scatter3d",
marker = list(size = 1, color = "black", symbol = 104)) %>%
add_trace(x = 1, y = 2/3, z = 1, mode = "markers", type = "scatter3d",
marker = list(size = 1, color = "black", symbol = 104)) %>%
add_trace(x = 1, y = -1/3, z = 1, mode = "markers", type = "scatter3d",
marker = list(size = 1, color = "black", symbol = 104)) %>%
add_trace(x = 1, y = -2/3, z = 1, mode = "markers", type = "scatter3d",
marker = list(size = 1, color = "gray", symbol = 104)) %>% # now the vextex above
add_trace(x = -1, y = 0, z = 1, mode = "markers", type = "scatter3d",
marker = list(size = 1, color = "gray", symbol = 104)) %>%
add_trace(x = 0, y = 0, z = 1, mode = "markers", type = "scatter3d",
marker = list(size = 1, color = "gray", symbol = 104)) %>%
add_trace(x = 0, y = 1, z = 1, mode = "markers", type = "scatter3d",
marker = list(size = 1, color = "gray", symbol = 104)) %>%
add_trace(x = 0, y = -1, z = 1, mode = "markers", type = "scatter3d",
marker = list(size = 1, color = "gray", symbol = 104)) %>%
add_trace(x = 1, y = 0, z = 1, mode = "markers", type = "scatter3d",
marker = list(size = 1, color = "gray", symbol = 104)) %>%
add_trace(x = 1, y = 1, z = 1, mode = "markers", type = "scatter3d",
marker = list(size = 1, color = "gray", symbol = 104)) %>%
add_trace(x = 1, y = -1, z = 1, mode = "markers", type = "scatter3d",
marker = list(size = 1, color = "gray", symbol = 104)) %>%
add_trace(data = df01, x = ~x, y = ~y, z = ~z, mode = "lines", type = "scatter3d",
line = list(width = 1, color = "gray", symbol = 104, dash = 'dot')) %>%
add_trace(data = df02, x = ~x, y = ~y, z = ~z, mode = "lines", type = "scatter3d",
line = list(width = 1, color = "gray", symbol = 104, dash = 'dot')) %>%
add_trace(data = df03, x = ~x, y = ~y, z = ~z, mode = "lines", type = "scatter3d",
line = list(width = 1, color = "gray", symbol = 104, dash = 'dot')) %>%
add_trace(data = df04, x = ~x, y = ~y, z = ~z, mode = "lines", type = "scatter3d",
line = list(width = 1, color = "gray", symbol = 104, dash = 'dot')) %>%
add_trace(data = df05, x = ~x, y = ~y, z = ~z, mode = "lines", type = "scatter3d",
line = list(width = 1, color = "gray", symbol = 104, dash = 'dot')) %>%
add_trace(data = df06, x = ~x, y = ~y, z = ~z, mode = "lines", type = "scatter3d",
line = list(width = 1, color = "gray", symbol = 104, dash = 'dot')) %>%
add_trace(data = df07, x = ~x, y = ~y, z = ~z, mode = "lines", type = "scatter3d",
line = list(width = 1, color = "gray", symbol = 104, dash = 'dot')) %>%
add_trace(data = df08, x = ~x, y = ~y, z = ~z, mode = "lines", type = "scatter3d",
line = list(width = 1, color = "gray", symbol = 104, dash = 'dot')) %>%
add_trace(data = df09, x = ~x, y = ~y, z = ~z, mode = "lines", type = "scatter3d",
line = list(width = 1, color = "gray", symbol = 104, dash = 'dot')) %>%
add_trace(data = df10, x = ~x, y = ~y, z = ~z, mode = "lines", type = "scatter3d",
line = list(width = 1, color = "gray", symbol = 104, dash = 'dot')) %>%
add_trace(data = df11, x = ~x, y = ~y, z = ~z, mode = "lines", type = "scatter3d",
line = list(width = 1, color = "gray", symbol = 104, dash = 'dot')) %>%
add_trace(data = df12, x = ~x, y = ~y, z = ~z, mode = "lines", type = "scatter3d",
line = list(width = 1, color = "gray", symbol = 104, dash = 'dot')) %>%
add_trace(data = df13, x = ~x, y = ~y, z = ~z, mode = "lines", type = "scatter3d",
line = list(width = 1, color = "gray", symbol = 104, dash = 'dot')) %>%
add_trace(data = df14, x = ~x, y = ~y, z = ~z, mode = "lines", type = "scatter3d",
line = list(width = 1, color = "gray", symbol = 104, dash = 'dot')) %>%
add_trace(data = df15, x = ~x, y = ~y, z = ~z, mode = "lines", type = "scatter3d",
line = list(width = 1, color = "gray", symbol = 104, dash = 'dot')) %>%
add_trace(data = df16, x = ~x, y = ~y, z = ~z, mode = "lines", type = "scatter3d",
line = list(width = 1, color = "gray", symbol = 104, dash = 'dot')) %>%
add_trace(data = df17, x = ~x, y = ~y, z = ~z, mode = "lines", type = "scatter3d",
line = list(width = 1, color = "gray", symbol = 104, dash = 'dot')) %>%
add_trace(data = df18, x = ~x, y = ~y, z = ~z, mode = "lines", type = "scatter3d",
line = list(width = 1, color = "gray", symbol = 104, dash = 'dot')) %>%
add_trace(data = df19, x = ~x, y = ~y, z = ~z, mode = "lines", type = "scatter3d",
line = list(width = 1, color = "gray", symbol = 104, dash = 'dot')) %>%
add_trace(data = df20, x = ~x, y = ~y, z = ~z, mode = "lines", type = "scatter3d",
line = list(width = 1, color = "gray", symbol = 104, dash = 'dot')) %>%
add_trace(data = df21, x = ~x, y = ~y, z = ~z, mode = "lines", type = "scatter3d",
line = list(width = 1, color = "gray", symbol = 104, dash = 'dot')) %>%
add_trace(data = df22, x = ~x, y = ~y, z = ~z, mode = "lines", type = "scatter3d",
line = list(width = 1, color = "gray", symbol = 104, dash = 'dot')) %>%
add_trace(data = df23, x = ~x, y = ~y, z = ~z, mode = "lines", type = "scatter3d",
line = list(width = 1, color = "gray", symbol = 104, dash = 'dot')) %>% #so far vertical
add_trace(data = d01, x = ~x, y = ~y, z = ~z, mode = "lines", type = "scatter3d",
line = list(width = 1, color = "gray", symbol = 104)) %>%
add_trace(data = d02, x = ~x, y = ~y, z = ~z, mode = "lines", type = "scatter3d",
line = list(width = 1, color = "gray", symbol = 104)) %>%
add_trace(data = d03, x = ~x, y = ~y, z = ~z, mode = "lines", type = "scatter3d",
line = list(width = 2, color = "red", symbol = 104)) %>%
add_trace(data = d04, x = ~x, y = ~y, z = ~z, mode = "lines", type = "scatter3d",
line = list(width = 1, color = "gray", symbol = 104)) %>%
add_trace(data = d05, x = ~x, y = ~y, z = ~z, mode = "lines", type = "scatter3d",
line = list(width = 1, color = "gray", symbol = 104)) %>%
add_trace(data = d06, x = ~x, y = ~y, z = ~z, mode = "lines", type = "scatter3d",
line = list(width = 1, color = "gray", symbol = 104)) %>%
add_trace(data = d07, x = ~x, y = ~y, z = ~z, mode = "lines", type = "scatter3d",
line = list(width = 1, color = "gray", symbol = 104)) %>%
add_trace(data = d08, x = ~x, y = ~y, z = ~z, mode = "lines", type = "scatter3d",
line = list(width = 1, color = "gray", symbol = 104)) %>%
add_trace(data = d09, x = ~x, y = ~y, z = ~z, mode = "lines", type = "scatter3d",
line = list(width = 2, color = "red", symbol = 104)) %>%
add_trace(data = d10, x = ~x, y = ~y, z = ~z, mode = "lines", type = "scatter3d",
line = list(width = 1, color = "gray", symbol = 104)) %>%
add_trace(data = d11, x = ~x, y = ~y, z = ~z, mode = "lines", type = "scatter3d",
line = list(width = 1, color = "gray", symbol = 104)) %>%
add_trace(data = d12, x = ~x, y = ~y, z = ~z, mode = "lines", type = "scatter3d",
line = list(width = 1, color = "gray", symbol = 104)) %>%
add_trace(data = d13, x = ~x, y = ~y, z = ~z, mode = "lines", type = "scatter3d",
line = list(width = 1, color = "gray", symbol = 104)) %>%
add_trace(data = d14, x = ~x, y = ~y, z = ~z, mode = "lines", type = "scatter3d",
line = list(width = 2, color = "blue", symbol = 104)) %>%
add_trace(data = d15, x = ~x, y = ~y, z = ~z, mode = "lines", type = "scatter3d",
line = list(width = 1, color = "gray", symbol = 104)) %>%
add_trace(data = d16, x = ~x, y = ~y, z = ~z, mode = "lines", type = "scatter3d",
line = list(width = 2, color = "blue", symbol = 104)) %>%
add_trace(data = d17, x = ~x, y = ~y, z = ~z, mode = "lines", type = "scatter3d",
line = list(width = 1, color = "gray", symbol = 104)) %>%
add_trace(data = d18, x = ~x, y = ~y, z = ~z, mode = "lines", type = "scatter3d",
line = list(width = 1, color = "gray", symbol = 104)) %>%
add_trace(data = d19, x = ~x, y = ~y, z = ~z, mode = "lines", type = "scatter3d",
line = list(width = 1, color = "gray", symbol = 104)) %>%
add_trace(data = d20, x = ~x, y = ~y, z = ~z, mode = "lines", type = "scatter3d",
line = list(width = 1, color = "gray", symbol = 104)) %>%
add_trace(data = d21, x = ~x, y = ~y, z = ~z, mode = "lines", type = "scatter3d",
line = list(width = 1, color = "gray", symbol = 104)) %>%
add_trace(data = d22, x = ~x, y = ~y, z = ~z, mode = "lines", type = "scatter3d",
line = list(width = 1, color = "gray", symbol = 104)) %>%
add_trace(data = d23, x = ~x, y = ~y, z = ~z, mode = "lines", type = "scatter3d",
line = list(width = 1, color = "gray", symbol = 104)) %>%
add_trace(data = d24, x = ~x, y = ~y, z = ~z, mode = "lines", type = "scatter3d",
line = list(width = 1, color = "gray", symbol = 104)) %>%
add_trace(data = d25, x = ~x, y = ~y, z = ~z, mode = "lines", type = "scatter3d",
line = list(width = 1, color = "gray", symbol = 104)) %>%
add_trace(data = d26, x = ~x, y = ~y, z = ~z, mode = "lines", type = "scatter3d",
line = list(width = 1, color = "gray", symbol = 104)) %>%
add_trace(data = d27, x = ~x, y = ~y, z = ~z, mode = "lines", type = "scatter3d",
line = list(width = 2, color = "red", symbol = 104)) %>%
add_trace(data = d28, x = ~x, y = ~y, z = ~z, mode = "lines", type = "scatter3d",
line = list(width = 1, color = "gray", symbol = 104)) %>%
add_trace(data = d29, x = ~x, y = ~y, z = ~z, mode = "lines", type = "scatter3d",
line = list(width = 1, color = "gray", symbol = 104)) %>%
add_trace(data = d30, x = ~x, y = ~y, z = ~z, mode = "lines", type = "scatter3d",
line = list(width = 1, color = "gray", symbol = 104)) %>%
add_trace(data = d31, x = ~x, y = ~y, z = ~z, mode = "lines", type = "scatter3d",
line = list(width = 1, color = "gray", symbol = 104)) %>%
add_trace(data = d32, x = ~x, y = ~y, z = ~z, mode = "lines", type = "scatter3d",
line = list(width = 1, color = "gray", symbol = 104)) %>%
add_trace(data = d33, x = ~x, y = ~y, z = ~z, mode = "lines", type = "scatter3d",
line = list(width = 2, color = "red", symbol = 104)) %>%
add_trace(data = d34, x = ~x, y = ~y, z = ~z, mode = "lines", type = "scatter3d",
line = list(width = 1, color = "gray", symbol = 104)) %>%
add_trace(data = d35, x = ~x, y = ~y, z = ~z, mode = "lines", type = "scatter3d",
line = list(width = 1, color = "gray", symbol = 104)) %>%
add_trace(data = d36, x = ~x, y = ~y, z = ~z, mode = "lines", type = "scatter3d",
line = list(width = 1, color = "gray", symbol = 104)) %>%
add_trace(data = d37, x = ~x, y = ~y, z = ~z, mode = "lines", type = "scatter3d",
line = list(width = 1, color = "gray", symbol = 104)) %>%
add_trace(data = d38, x = ~x, y = ~y, z = ~z, mode = "lines", type = "scatter3d",
line = list(width = 1, color = "gray", symbol = 104)) %>%
add_trace(data = d39, x = ~x, y = ~y, z = ~z, mode = "lines", type = "scatter3d",
line = list(width = 1, color = "gray", symbol = 104)) %>%
add_trace(data = d40, x = ~x, y = ~y, z = ~z, mode = "lines", type = "scatter3d",
line = list(width = 1, color = "gray", symbol = 104)) %>%
add_trace(data = d41, x = ~x, y = ~y, z = ~z, mode = "lines", type = "scatter3d",
line = list(width = 1, color = "gray", symbol = 104)) %>%
add_trace(data = d42, x = ~x, y = ~y, z = ~z, mode = "lines", type = "scatter3d",
line = list(width = 1, color = "gray", symbol = 104)) %>%
add_trace(data = d43, x = ~x, y = ~y, z = ~z, mode = "lines", type = "scatter3d",
line = list(width = 1, color = "gray", symbol = 104)) %>%
add_trace(data = d44, x = ~x, y = ~y, z = ~z, mode = "lines", type = "scatter3d",
line = list(width = 1, color = "gray", symbol = 104)) %>%
add_trace(data = d45, x = ~x, y = ~y, z = ~z, mode = "lines", type = "scatter3d",
line = list(width = 1, color = "gray", symbol = 104)) %>%
add_trace(data = d46, x = ~x, y = ~y, z = ~z, mode = "lines", type = "scatter3d",
line = list(width = 1, color = "gray", symbol = 104)) %>%
add_trace(data = d47, x = ~x, y = ~y, z = ~z, mode = "lines", type = "scatter3d",
line = list(width = 1, color = "gray", symbol = 104)) %>%
add_trace(data = d48, x = ~x, y = ~y, z = ~z, mode = "lines", type = "scatter3d",
line = list(width = 1, color = "gray", symbol = 104)) %>% #star set
add_trace(data = d49, x = ~x, y = ~y, z = ~z, mode = "lines", type = "scatter3d",
line = list(width = 3, color = "green", symbol = 104)) %>%
add_trace(data = d50, x = ~x, y = ~y, z = ~z, mode = "lines", type = "scatter3d",
line = list(width = 3, color = "green", symbol = 104)) %>%
add_trace(data = d51, x = ~x, y = ~y, z = ~z, mode = "lines", type = "scatter3d",
line = list(width = 3, color = "green", symbol = 104)) %>%
add_trace(data = d52, x = ~x, y = ~y, z = ~z, mode = "lines", type = "scatter3d",
line = list(width = 3, color = "green", symbol = 104))Figure 3. Illustration of the system of basis functions \(\{\varphi_j^e, \phi_v\}\) (solid gray lines) on the graph \(\Gamma= (\mathcal{V},\mathcal{E})\), where \(\mathcal{E}= \left\{e_1,\dots,e_8\right\}\) and \(\mathcal{V}= \left\{v_1,\dots,v_7\right\}\). Note that for all \(e_i\in\mathcal{E}\), \(n_{e_i} = 3\) and \(h_{e_i} = 1/3\). In particular, for edge \(e_7\), observe that the three regular segments are delimited by the nodes \(v_6 = x_0^{e_7}, x_1^{e_7}, x_2^{e_7}, x_3^{e_7} = v_7\). Corresponding to node \(x_1^{e_7}\), we have plotted the basis function \(\varphi_1^{e_7}\) in blue. Notice also that the set \(\mathcal{N}_{v_2}\) is depicted in green and its corresponding basis function \(\phi_{v_2}\) is shown in red.
6 Basis functions, alternative illustrations
V <- rbind(c(0, 0), c(1, 0), c(1, 1), c(0, 1), c(-1, 1), c(-1, 0), c(0, -1))
E <- rbind(c(1, 2), c(2, 3), c(3, 4), c(4, 5),
c(5, 6), c(6, 1), c(4, 1),c(1, 7))
graph <- metric_graph$new(V = V, E = E)
graph_to_get_loc <- graph$clone()
graph_to_get_loc$build_mesh(h = 1/150)
graph_to_get_loc$plot(mesh = TRUE) +
ggtitle("Fine mesh") +
theme_minimal() +
theme(text = element_text(family = "Palatino"))
loc <- graph_to_get_loc$get_mesh_locations()
graph$build_mesh(h = 1/3)
graph$plot(mesh = TRUE) +
ggtitle("Coarse mesh") +
theme_minimal() +
theme(text = element_text(family = "Palatino"))
A <- graph$fem_basis(loc)
DD <- data.frame(A = as.matrix(A)) %>%
rowwise() %>%
mutate(max_value = max(c_across(everything()))) %>%
cbind(edge_number = loc[, 1], distance_on_edge = loc[, 2], A.24 = 0)
graph$add_observations(data = DD, normalized = TRUE)graph$plot(data = "max_value", vertex_size = 0, plotly = TRUE, edge_width = 0.1) %>%
config(mathjax = 'cdn') %>%
layout(title = TeX("\\text{Basis functions on a metric graph }\\Gamma = (\\mathcal{V},\\mathcal{E})"),
showlegend = FALSE,
font = list(family = "Palatino"),
scene = list(aspectratio = list(x = 1.8, y = 1.8, z = 0.4)))graph$plot(data = "A.1", vertex_size = 0, plotly = TRUE, edge_width = 0.1) %>%
config(mathjax = 'cdn') %>%
layout(title = TeX("\\text{One basis function on a metric graph }\\Gamma = (\\mathcal{V},\\mathcal{E})"),
showlegend = FALSE,
font = list(family = "Palatino"),
scene = list(aspectratio = list(x = 1.8, y = 1.8, z = 0.4)))graph_to_get_loc$plot_function(X = DD$max_value, vertex_size = 0, plotly = TRUE) %>%
config(mathjax = 'cdn') %>%
layout(title = TeX("\\text{Basis functions on a metric graph }\\Gamma = (\\mathcal{V},\\mathcal{E})"),
showlegend = FALSE,
font = list(family = "Palatino"),
scene = list(aspectratio = list(x = 1.8, y = 1.8, z = 0.4)))graph_to_get_loc$plot_function(X = DD$A.1, vertex_size = 0, plotly = TRUE) %>%
config(mathjax = 'cdn') %>%
layout(title = TeX("\\text{One basis function on a metric graph }\\Gamma = (\\mathcal{V},\\mathcal{E})"),
showlegend = FALSE,
font = list(family = "Palatino"),
scene = list(aspectratio = list(x = 1.8, y = 1.8, z = 0.4)))# Initialize the first plot
p1 <- graph_to_get_loc$plot_function(X = DD[[paste0("A.", 1)]], line_color = "red",
edge_width = 2, line_width = 2, support_width = 0.2, plotly = TRUE)
# Loop through to create the remaining plots
for (i in 2:24) {
line_color <- "gray"
if (i == 11) {line_color <- "blue"}
if (i == 24) {line_color <- "black"}
X <- DD[[paste0("A.", i)]]
p <- get(paste0("p", i - 1))
assign(paste0("p", i), graph_to_get_loc$plot_function(X = X, line_color = line_color, edge_width = 2, line_width = 2, support_width = 0.2, plotly = TRUE, p = p))
}
pp = p24 %>%
config(mathjax = 'cdn') %>%
layout(title = TeX("\\text{Basis functions on a metric graph }\\Gamma = (\\mathcal{V},\\mathcal{E})"),
showlegend = FALSE,
font = list(family = "Palatino"),
scene = list(
aspectratio = list(x = 1.8, y = 1.8, z = 0.4),
annotations = list(
list(
x = -1, y = 0, z = 0,
text = TeX("v_1"),
textangle = 0, ax = 0, ay = 25,
font = list(color = "black", size = 16),
arrowcolor = "black", arrowsize = 1, arrowwidth = 1,arrowhead = 1),
list(
x = 0, y = 0, z = 0,
text = TeX("v_2"),
textangle = 0, ax = 0, ay = 25,
font = list(color = "black", size = 16),
arrowcolor = "black", arrowsize = 1, arrowwidth = 1,arrowhead = 1),
list(
x = 0, y = -1, z = 0,
text = TeX("v_3"),
textangle = 0, ax = 0, ay = 25,
font = list(color = "black", size = 16),
arrowcolor = "black", arrowsize = 1, arrowwidth = 1,arrowhead = 1),
list(
x = 1, y = -1, z = 0,
text = TeX("v_4"),
textangle = 0, ax = 0, ay = 25,
font = list(color = "black", size = 16),
arrowcolor = "black", arrowsize = 1, arrowwidth = 1,arrowhead = 1),
list(
x = 1, y = 0, z = 0,
text = TeX("v_5"),
textangle = 0, ax = 0, ay = 25,
font = list(color = "black", size = 16),
arrowcolor = "black", arrowsize = 1, arrowwidth = 1,arrowhead = 1),
list(
x = 1, y = 1, z = 0,
text = TeX("v_6,x_0^{e_7}"),
textangle = 0, ax = 0, ay = 25,
font = list(color = "black", size = 16),
arrowcolor = "black", arrowsize = 1, arrowwidth = 1,arrowhead = 1),
list(
x = 0, y = 1, z = 0,
text = TeX("v_7,x_3^{e_7}"),
textangle = 0, ax = 0, ay = 25,
font = list(color = "black", size = 16),
arrowcolor = "black", arrowsize = 1, arrowwidth = 1,arrowhead = 1),
list(
x = 2/3, y = 1, z = 0,
text = TeX("x_1^{e_7}"),
textangle = 0, ax = 0, ay = 25,
font = list(color = "black", size = 16),
arrowcolor = "black", arrowsize = 1, arrowwidth = 1,arrowhead = 1),
list(
x = 1/3, y = 1, z = 0,
text = TeX("x_2^{e_7}"),
textangle = 0, ax = 0, ay = 25,
font = list(color = "black", size = 16),
arrowcolor = "black", arrowsize = 1, arrowwidth = 1,arrowhead = 1),
list(
x = 2/3, y = 1, z = 1,
text = TeX("\\varphi_1^{e_7}"),
textangle = 0, ax = 0, ay = 75,
font = list(color = "blue", size = 16),
arrowcolor = "white", arrowsize = 1, arrowwidth = 1,arrowhead = 1),
list(
x = 0, y = 0, z = 1,
text = TeX("\\phi_{v_2}"),
textangle = 0, ax = 0, ay = 75,
font = list(color = "red", size = 16),
arrowcolor = "white", arrowsize = 1, arrowwidth = 1,arrowhead = 1)
))) %>%
add_trace(x = 0, y = 0, z = 0, mode = "markers", type = "scatter3d",
marker = list(size = 4, color = "blue", symbol = 104)) %>%
add_trace(x = 0, y = 1, z = 0, mode = "markers", type = "scatter3d",
marker = list(size = 4, color = "blue", symbol = 104)) %>%
add_trace(x = 1, y = 1, z = 0, mode = "markers", type = "scatter3d",
marker = list(size = 4, color = "blue", symbol = 104)) %>%
add_trace(x = 1, y = 0, z = 0, mode = "markers", type = "scatter3d",
marker = list(size = 4, color = "blue", symbol = 104)) %>%
add_trace(x = 1, y = -1, z = 0, mode = "markers", type = "scatter3d",
marker = list(size = 4, color = "blue", symbol = 104)) %>%
add_trace(x = 0, y = -1, z = 0, mode = "markers", type = "scatter3d",
marker = list(size = 4, color = "blue", symbol = 104)) %>%
add_trace(x = -1, y = 0, z = 0, mode = "markers", type = "scatter3d",
marker = list(size = 4, color = "blue", symbol = 104)) %>%
add_trace(x = -2/3, y = 0, z = 0, mode = "markers", type = "scatter3d", # before this
marker = list(size = 4, color = "blue", symbol = 104)) %>%
add_trace(x = -1/3, y = 0, z = 0, mode = "markers", type = "scatter3d",
marker = list(size = 4, color = "blue", symbol = 104)) %>%
add_trace(x = 0, y = -1/3, z = 0, mode = "markers", type = "scatter3d",
marker = list(size = 4, color = "blue", symbol = 104)) %>%
add_trace(x = 0, y = -2/3, z = 0, mode = "markers", type = "scatter3d",
marker = list(size = 4, color = "blue", symbol = 104)) %>%
add_trace(x = 1/3, y = -1, z = 0, mode = "markers", type = "scatter3d",
marker = list(size = 4, color = "blue", symbol = 104)) %>%
add_trace(x = 2/3, y = -1, z = 0, mode = "markers", type = "scatter3d",
marker = list(size = 4, color = "blue", symbol = 104)) %>%
add_trace(x = 0, y = 1/3, z = 0, mode = "markers", type = "scatter3d",
marker = list(size = 4, color = "blue", symbol = 104)) %>%
add_trace(x = 0, y = 2/3, z = 0, mode = "markers", type = "scatter3d",
marker = list(size = 4, color = "blue", symbol = 104)) %>%
add_trace(x = 1/3, y = 0, z = 0, mode = "markers", type = "scatter3d",
marker = list(size = 4, color = "blue", symbol = 104)) %>%
add_trace(x = 2/3, y = 0, z = 0, mode = "markers", type = "scatter3d",
marker = list(size = 4, color = "blue", symbol = 104)) %>%
add_trace(x = 1/3, y = 1, z = 0, mode = "markers", type = "scatter3d",
marker = list(size = 4, color = "blue", symbol = 104)) %>%
add_trace(x = 2/3, y = 1, z = 0, mode = "markers", type = "scatter3d",
marker = list(size = 4, color = "blue", symbol = 104)) %>%
add_trace(x = 1, y = 1/3, z = 0, mode = "markers", type = "scatter3d",
marker = list(size = 4, color = "blue", symbol = 104)) %>%
add_trace(x = 1, y = 2/3, z = 0, mode = "markers", type = "scatter3d",
marker = list(size = 4, color = "blue", symbol = 104)) %>%
add_trace(x = 1, y = -1/3, z = 0, mode = "markers", type = "scatter3d",
marker = list(size = 4, color = "blue", symbol = 104)) %>%
add_trace(x = 1, y = -2/3, z = 0, mode = "markers", type = "scatter3d",
marker = list(size = 4, color = "blue", symbol = 104)) %>%
add_trace(data = d49, x = ~x, y = ~y, z = ~z, mode = "lines", type = "scatter3d",
line = list(width = 3, color = "green", symbol = 104)) %>%
add_trace(data = d50, x = ~x, y = ~y, z = ~z, mode = "lines", type = "scatter3d",
line = list(width = 3, color = "green", symbol = 104)) %>%
add_trace(data = d51, x = ~x, y = ~y, z = ~z, mode = "lines", type = "scatter3d",
line = list(width = 3, color = "green", symbol = 104)) %>%
add_trace(data = d52, x = ~x, y = ~y, z = ~z, mode = "lines", type = "scatter3d",
line = list(width = 3, color = "green", symbol = 104))